{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]], and let $C(X)$ be the algebra of [[continuous|0X3WGX95FE]] functions $X \\to \\C$ with the sup norm $\\norm{f}_\\infty = \\max_{x \\in X} \\abs{f(x)}$.\nA *uniform algebra* on $X$ is a subalgebra $A \\subseteq C(X)$ such that:\n\n1. $A$ is closed in $C(X)$ for the sup norm, that is, under [[uniform convergence|08VZ57PT26]];\n2. $A$ contains the constant functions;\n3. $A$ separates the points of $X$: for $x \\ne y$ in $X$ there is $f \\in A$ with $f(x) \\ne f(y)$.\n\n## Remarks\n\nA uniform algebra is a commutative unital Banach algebra with $\\norm{f^2} = \\norm{f}^2$ for all $f$.\nConversely, a commutative unital complex Banach algebra with $\\norm{a^2} = \\norm{a}^2$ for all $a$ is isometrically isomorphic, through the Gelfand transform, to a uniform algebra on its character space.\n\nExample: the disc algebra $A(\\mathbb{D})$ on the closed unit disc $X = \\overline{\\mathbb{D}}$, the continuous functions on $X$ that are holomorphic in the open disc.\nIt is a subalgebra; it is closed because [[uniform limits of holomorphic functions are holomorphic|0431RN0R1N]]; it contains the constants; and $f(z) = z$ separates points.\n\nNon-example: on $X = [0, 1]$, let $A = \\set{f \\in C([0,1]) : f(0) = f(1)}$.\nIt is a closed subalgebra containing the constants, but no $f \\in A$ distinguishes $0$ from $1$, so clause 3 fails.\n\nBy the [[Stone–Weierstrass theorem|1P8D3MK79H]], a uniform algebra on $X$ that is closed under complex conjugation equals $C(X)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "ZP55RF3A0C",
    "kind": "definition",
    "msc": [
      "46J10",
      "46J05"
    ],
    "name": "Uniform algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Uniform_algebra"
      }
    ],
    "tag": "D5CKZCK3Z2",
    "uses": [
      "0TVJZYB6SH",
      "0S1FJ8VEX9",
      "0X3WGX95FE",
      "08VZ57PT26"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/D5CKZCK3Z2/",
  "links": [
    {
      "field": "defines",
      "target": "ZP55RF3A0C"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "0S1FJ8VEX9"
    },
    {
      "field": "uses",
      "target": "0X3WGX95FE"
    },
    {
      "field": "uses",
      "target": "08VZ57PT26"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/D5CKZCK3Z2.md",
  "msc": [
    "46J10",
    "46J05"
  ],
  "name": "Uniform algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D5C/D5CKZCK3Z2.md",
  "status": null,
  "tag": "D5CKZCK3Z2",
  "uri": "m2://D5CKZCK3Z2",
  "url": "https://math2.tamnd.com/wiki/uniform-algebra-definition/",
  "writing": {
    "tag": "D5CKZCK3Z2",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}